Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 2 x + 3 x^{2} )( 1 - 2 x + 4 x^{2} - 6 x^{3} + 9 x^{4} )$ |
| $1 - 4 x + 11 x^{2} - 20 x^{3} + 33 x^{4} - 36 x^{5} + 27 x^{6}$ | |
| Frobenius angles: | $\pm0.210767374595$, $\pm0.304086723985$, $\pm0.567777800232$ |
| Angle rank: | $3$ (numerical) |
| Jacobians: | $2$ |
| Isomorphism classes: | 8 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $12$ | $1584$ | $26676$ | $658944$ | $18035292$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $0$ | $16$ | $36$ | $100$ | $300$ | $736$ | $2016$ | $6460$ | $19872$ | $58816$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which 1 is hyperelliptic), and hence is principally polarizable:
- $y^2=2x^8+x^7+x^4+2x+2$
- $2x^4+2x^3y+x^3z+x^2yz+y^4+z^4=0$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3}$.
Endomorphism algebra over $\F_{3}$| The isogeny class factors as 1.3.ac $\times$ 2.3.ac_e and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 3.3.a_d_ae | $2$ | 3.9.g_bb_do |
| 3.3.a_d_e | $2$ | 3.9.g_bb_do |
| 3.3.e_l_u | $2$ | 3.9.g_bb_do |